Comments

  • A -> not-A
    "See the “⊢ Q” at the end? That means that Q follows from the bit before." Okay; can you spell it out for me? It's still not clicking.
  • A -> not-A
    P5. (P ∨ Q) ∧ ¬P ⊢ Q (disjunctive syllogism) I do not understand the move from P5 to C1 using disjunctive syllogism. Would you mind explaining?
  • A -> not-A


    " ¬∃x(P∧Q) "

    where x is an interpretation, P is "all premises are true" and Q is "the conclusion is false."

    Is there something problematic about writing the definition of validity that way?
  • A -> not-A
    By your own definition the argument is not valid.
  • A -> not-A
    If the first premise were agreed to, that would mean the disjunctive elimination leading to C1 would not work. If P and not-P are accepted, I take it that they are accepted propositions throught the entire proof. Unless P is suddenly not accepted in P5?
  • A -> not-A
    Forget "formal axiomatic system," a contradictory argument is always a problem. The "principle" of explosion directly infringes the law of non-contradiction. It's silly to even call it a principle.
  • A -> not-A
    The wikipedia article you cited literally says the principle of explosion is "disastrous" and "trivializes truth and falsity."
  • A -> not-A
    Ah, I see, then we will say as a shorthand "invalid" as a way of saying it does not follow, that is, that the conclusion cannot be derived using a priori reasoning.

    My question is, if I use a priori reasoning, how can I conclude that "I live in Antartica" (assuming that is true) based on the premise "Pluto is a planet and Pluto is not a planet". How does the conclusion "follow?" I saw your reasoning from the earlier argument, I'm just wondering what rule of inference leads to this conclusion.

    To be more specific, it seems to me that in the argument you stated, P1, P5, and C1 cannot all be true. That is, if C1 is true then P1 cannot be true. And if P1 is true then C1 cannot be.
  • A -> not-A
    "You can use the rules of inference to derive the conclusion "I am mortal" using a priori reasoning, but you cannot use the rules of inference to derive the conclusion "I am English" using a priori reasoning"

    That is well said.

    Perhaps we we disagree about what may be considered a rule of inference. Unless you think an argument that is invalid only coincidentally doesn't follow? Or is it invalid because it does not follow?
  • A -> not-A
    Okay I agree with you that only one of those two arguments is valid. Now, in a non-circular way, explain why the one follows but the other does not.
  • A -> not-A
    Why not? It satisfies the definition, does it not?
  • A -> not-A
    If I did live in Antartica it would have to be valid wouldn't it?
  • A -> not-A
    Your argument is that: If logicians have defined validity, then that definition is correct. Logicians have defined validity. Therefore, that definition is correct. This is a valid argument as far as I can tell. It is, however, unsound, as premise 1 is faulty.
  • A -> not-A
    Besides, if someone gave the argument you gave -- "I am a man and I am not a man. Therefore I am rich" that is a nonsensical argument; the conclusion just has nothing to do with the premises, you might as well argue "I am a human and it might snow this week, therefore I live in Antartica." Even if conclusion and premise are all true i.e. the argument is sound, what kind of argument is that?
  • A -> not-A
    It seems that that argument would be valid, but only if one accepts that an argument is valid iff there is no interpretation s.t. all premises are true and the conclusion is false per Tones' definition.

    If it turned out that validity required more than what that definition suggests (I think it does), then the argument you stated may well turn out to not be valid, as I think is the case.

    Maybe another way of coming at this is as follows - the conclusion is true. Period. Under that understanding, "there is no interpretation where the conclusion is false" ergo there is no interpretation s.t. all the premises are true and the conclusion is false. But the conclusion being true does not seem to guarantee that the argument is valid. But with Tones' definition, it would. Similarly, inconsistent premises also guarantee the validity of the argument according to Tones' definition, but that also seems problematic.
  • A -> not-A
    "Validity has to do with the conclusion following from the premises, and inconsistency is not evidence that the conclusion follows from the premises."Leontiskos

    That ((P→Q)∧Q), therefore P is not valid, whereas ((A∧¬A)∧(P→Q)∧Q), therefore P is valid, does seem strange to me. Inconsistent premises don't seem to have anything to do with whether the argument "follows." Although I have a feeling that Tones will have something to say about that.
  • A -> not-A
    One of the main takeaways from this discussion, for me, is that while some formal arguments may be valid, they are not necessarily valid in an informal setting.

    To wit,

    B
    Therefore A→B
    Formally valid.

    Water was added to the lake.
    Therefore,
    If it is cloudy out, then water was added to the lake.
    Informally not valid.

    as well as -

    A ^ B
    Therefore, (A→B).
    Formally valid.

    Kangaroos are marsupials and Paris is the capital of France.
    Therefore,
    If kangaroos are marsupials, then Paris is the capital of France.
    Informally not valid.
  • A -> not-A
    If I am referring to the right quotation, you said:
    No, it doesn't result in a contradiction. The conclusion is ~A, which is not a contradiction. Yes, the premises are inconsistent, but your definition of "rule" doesn't disallow inconsistent sets of premises, only required is that application of the rule doesn't allow a conclusion that is a contradiction. The particular application you mentioned doesn't derive a contradiction.TonesInDeepFreeze

    What I responded with --a rule must have been "followed" not merely be "present" and the use of a rule may not result in a contradiction means that the use of a rule, or I guess you would call it an operator or connective, whatever you call it, must not result in a contradiction. A->not-A, when this rule is applied and followed, that is, when it is true that "A" and the rule "A->not-A" is actually applied, a contradiction results, specifically "A and not-A."

    By "actually applied" I mean that the rule, or connective, does work in leading to the conclusion.

    The "following" of a rule versus it's being merely "present" can be illustrated by the following example:
    A->B
    B^C
    Therefore, C.
    In this example, the rule A-> B does not do any work, so even if it did result in a contradiction, the fact that it doesn't do any work in the argument and isn't followed or actually applied, means that the argument could still be valid.
  • A -> not-A
    Down the slippery slope of formalized illogicality.
  • A -> not-A
    Then note:

    P -> Q |= ~P v Q
    and
    ~P v Q |= ~P v Q
    TonesInDeepFreeze

    I think you meant:

    P -> Q |= ~P v Q
    and
    ~P v Q |= P -> Q

    ?
  • A -> not-A
    Not just conjunction, no, but having the same truth functionality as conjunction yes, just meta-logically different (if I am using that terminology correctly).
  • A -> not-A
    No, I read it, I just think you're disregarding the proviso I stated, namely that a rule must actually have been followed, not merely be present in an argument.

    As for the instantiation of truth possibilities by the rules, what I mean is that the possibilities for what is true and what is false are arrayed across a truth table. The rules must account for all the ways that those truth possibilities can be instantiated. So for the expression A v B, the truth table is T, T, T, F. On the other hand, T, F, F, F, is A ^ B. Every possibility wherein T is present must be uniquely accounted for by the rules. So T, F, F, F, and F, T, F, F, and F, F, T, F, and F, F, F, T, must all be "achievable instantiations" based on the rules we bring to the variables. If A v B were the only rule we applied, then not all of the truth possibilities could be instantiated, does that answer?
  • A -> not-A
    By "the following of a rule" I mean a literal rule such as a connective is actually used to reach a conclusion. The argument A->not-A therefore not-A does not, in my opinion, make any use of the conditional such that any rule has been followed. With the argument A->not-A, A, therefore not-A, the following of the rule, namely the conditional in that argument, leads to a contradiction between A and not-A, as such, it is disqualified from being a valid argument according to my definition.
  • A -> not-A
    According to you, what is the full meaning of P -> Q?TonesInDeepFreeze

    I may have mispoken, but to me the full meaning of "If P then Q" captures the fact that "P does not imply Q" can still be true even though not-P v Q can still be true. But then I now think P->Q is a meaningless expression so saying it "means" the same think as not-P or Q is unsubstantiated.
  • A -> not-A
    Whether or not the two expressions are semantically equivalent in a meta-logical sense depends on how one is using them.Leontiskos

    Hmm interesting, I think my position is that the formal conditional is meaningless then, insofar as it is just symbol manipulation.

    You could say that, but you would end up having to admit that "P does not imply Q" cannot be formalized in any way whatsoever, at least in propositional logic.Leontiskos

    I have tried to formalize it and can't seem to do so; this is an approximation:

    (A v ~A) → (~B v ~A)

    When (B and A) are both true, the expression seems to be false. On the other hand, the negation of that expression seems to imply that (A and B) must both be true. If the conditional is construed as only being true when A and B are true, then the negation of the initial expression maps onto A→B. Perhaps that could be written as, it is not the case that A does not imply B therefore A implies B. (Though if that were the case then A→B would be logically equivalent to A^B, although not meta-logically equivalent).

    But then I don't mind saying "P does not Imply Q" can't be formalized.
  • A -> not-A
    I get mixed up with this, but I think the disjunction (not-P or Q) can still be true even if P does not imply Q. So the "meaning" of the disjunctive is not specific enough.
  • A -> not-A
    It seems to me that the disjunctive equivalent does not capture the full meaning of P->Q.
  • A -> not-A
    You can absolutely substitute them logically, however I do not think they mean the same thing. P->Q either means just that "P->Q" or it doesn't have a meaning at all, either way P->Q does not, in my opinion, mean the same thing as its logical equivalent.
  • A -> not-A
    1. Right, I mean P entails Q. The logical equivalence (not-P or Q) is an implication of the conditional, not having the same meaning as the conditional.

    2. I take your question to be what would a rule be, how is it defined? I would define a rule as a member belonging to a set that exhausts all "truth possibilities." I would add that the following of a rule may not result in a contradiction.

    A rule relating two different variables would have (I think) 15 possible truth configurations. The rules must at least enable all those possibilities to be instantiated (though perhaps it may exclude possibilities that are necessarily contradictory).

    3. "Some proposition is not the case"
    Both propositions must be true
    Either proposition must be true
    If the one proposition is true, so must the consequent proposition
    Both propositions are either both true or both false.

    5. Valid argument = following the rules, where rules are defined as those operations that enable each truth possibility to be instantiated but that do not result in a contradiction by following that rule.

    8. Not logical anarchy; the rules must enable all truth possibilities to be instantiated except that the rule may not result in a contradiction if it is followed.

    This way of defining validity may be preferable because it deals with cases such as A->not-A therefore Not-A that are intuitively illogical; such an argument does not involve the following of a rule, and so it is not valid.

    Similarly, A, A->not-A therefore not-A another intuitively illogical seeming argument would not be valid because the following of the rule results in a contradiction.
  • A -> not-A
    1. I take a conditional to be saying: if the antecedent is true, it can't be the case (there is no circumstances such) that the consequent is false.

    2. Rather than a correct conclusion, all we need are conclusions that follow the relevant rules, any and all such conclusions are legitimate.

    3. I refer to connectives as rules.

    4. Then we are out of luck.

    5. I drop the truth preservation condition for validity.

    8. If we drop the truth preservation part of the definition, it is not circular. An argument is valid where it follows the relevant rules. Period. I don't think it is necessary for me to stipulate that a rule be followed "correctly," just that it be followed.
  • A -> not-A
    1. The conditional means that in the event that the antecedent is true, the consequent must be true. It is one of the logical rules that must be followed for the argument to be valid.

    2. Provided that a set of all conclusions follows the rules correctly and is exhaustive of all such conclusions, that set encompasses all legitimate conclusions.

    3. logical operators.

    4. We would have to ask the speaker to clarify.

    5. Noted, let's set aside questions concerning meaning; the second definition may have more problems then I can resolve.

    6. Okay.

    7. So then "the truth of the premises guarantees the truth of the conclusion" is the same as "there is no interpretation (assignment of truth values) such that the premises are all true and the conclusion is false" ?

    8. So I think what I am trying to say is that the definition of validity is following the rules correctly. And that following the rules correctly is defined by rule-following that results in truth preservation. Such that, truth preservation is a consequence of rule following, and it is the rule following itself that is responsible for the validity. In other words, the premises themselves don't guarantee the truth of the conclusion, rather the following of the rule(s), given that the premises are all true, is what guarantees the truth of the conclusion. Put another way, truth preservation does not make the argument rule-following, but rule-following is what makes the argument truth preserving. (Truth preservation does make the rule-following "correct.") Not sure if that totally makes sense.
  • A -> not-A
    So actually, I would say my definition of valid is different from the ordinary formal logic definition in that I am defining validity in terms of rule-following, not in terms of truth-preservation; truth-preservation is more like a consequence of the definition.
  • A -> not-A
    P->Q. P. Therefore, not-Q. would both flout the meaning of the conditional, and in such a way that it changes the conclusion. It's different than what the conclusion should be (namely Q).

    I don't understand the second question.

    Third question answered as correctly used rules is defined.

    I don't know the difference between propositional logic and ordinary formal logic so I do not know how to answer this one.

    The meaning of an expression depends on what the speaker intended by it - natural language I would think would go along way in dissolving confusion over what is meant.

    Right, where there is disagreement over a meaning, that meaning is not well-formed and not suitable for logical operations. I would expect something like that to be true for your definition of validity as well.

    I guess I agree with the ordinary definition of valid in formal logic. That is not the definition you cited earlier in the thread - the definition that I am suggesting an alternative to.

    I do not see truth preservation as synonymous with validity; I defined validity as rule following; a rule is followed correctly if it preserves truth; I didn't define validity as truth-preserving. Truth preservation is a consequence of validity, namely, following the relevant rules correctly.
  • A -> not-A
    And a relevant rule is correctly followed just in case.. if it were the case that all the premises were true and the relevant rule is followed, then the conclusion must also be true.
  • A -> not-A
    Relevant rules like conditionals "And" "Or" operators-- when those are used correctly the rules are followed and the argument may be considered valid. Any rule that is such that if it weren't followed, the conclusion would be different, is a relevant rule. The rules would ideally be universal and based on logical intuition; if people use different sets of rules, then the rules must be clearly communicated so that that "logic" can be understood or followed.

    The meaning of the premise and conclusion depends on the expressions used (I guess this definition isn't unequivocal as it would only apply to ordinary natural language, not to formal logic). I don't know any theories of meaning so I can't answer that. If the meanings differ, then I'm not really sure what the result would be, seems like communication is out the door let alone logic if we can't agree on the same meaning of words and sentences.
  • A -> not-A
    Here are two ideas for defining validity: (1) an argument is valid when all the relevant rules are followed. Or, (2) an argument is valid when the meaning of the premises leads to the meaning of the conclusion.
  • A -> not-A
    It has been a long time since I learned some logic and I wasn't great at it, but I do know what truth tables are and I think how to use them.; I don't see how that implies a definition of "validity" using classical logic.
  • A -> not-A
    I mean take the definition of validity, and write it as an expression using symbols and logical operators; is that something that can be done?

    I don't mean examples of valid arguments, I am referring to the definition itself.
  • A -> not-A
    Okay, I actually do get that the example I just gave has "an interpretation wherein all the premises are true and the conclusion is false" such that it is "not valid." " Would you care to formalize the validity definition as it concerns arguments and do so using logical operators? I was trying to apply De Morgan's laws to your definition but I don't think it worked. On a side note, Banno I can hear your laughter and it is most unwelcome at this time.
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